Linux, Save As Game Engine Runtime not working
Which exporter addon did you use? If you used the 'save game as runtime' it creates an executable for the platform you are on. This is found in the File->Export->Save Game as Runtime

If you use the 'Game Engine Publishing', you also need to download blender for the different OS's, and point the addon at the filepath for blender. This is a fair bit more complex, and the options are found in the Render Panel.



I'm going to assume you're using the 'save game as runtime'
If you are running windows, it will create an exe, if you are on linux it will create a linux executable - which does not have a file extension.
In a quick test I did, exporting it from my system worked find, although it took several seconds.

One thing to check is that there is a located blenderplayer file. When you chose 'save game as runtime,' you see this on the left:

Check that there is a file at the location in the circled box.

Also check what folder you're exporting it too. Neither of the folders shown in the question have the exported executable.